# Verification of the fixed-double-pole degree calculation

The eight-page manuscript gives complete proofs of its stated all-parameter
theorems over the complex numbers. This is an originating-researcher self-audit,
not independent peer review or proof-assistant certification. The broad AIM
source question is not resolved in its entirety.

## Proof obligations checked

Generic coordinate lines compute the two actual partial degrees after
cancellation. Every finite zero or pole is assigned to a chronological birth,
and the alternating simple poles at infinity are retained. In the pure and
QRT cases the finite packet ends in a unit; a later birth is counted afresh.
The nonresonant case excludes exactly nontrivial roots of unity of `-delta^3`,
not the value one.

In the mixed resonant case the surviving simple-pole coefficient is derived
including subleading terms. The three required exceptional coefficients are
excluded using an algebraic-integer obstruction, a cyclotomic factorization
and a resultant equal to 27. The constant-multiplicity affine orbit and the
primitive-cubic three-block return are treated separately.

An analytic product chart proves the repeated higher-order cancellations.
Only unit denominators are inverted, and the last zero is not inverted during
the return. Dominance of the first-birth divisor makes its predecessor residue
transcendental, preventing the first-return leading coefficient from vanishing.
The chart then proves the linearly growing multiplicities at every subsequent
return. The two multiplicity patterns yield the same zero/pole kernel ratio,
and hence the claimed rational generating function.

The positive-root argument uses exact polynomial identities, uniqueness of
the positive root, Pringsheim's theorem and existence of the dynamical degree
for birational surface maps. It is not inferred from a finite fitted sequence.

## Exact regression and reproducibility

The portable scripts use exact rational and finite-field arithmetic. The
nonresonant checker tests 198 partial-degree values and negative controls.
The pure-resonance checker tests 240 partial-degree values, including both
axes and several resonance orders. The mixed checker tests 160 values,
including both multiplicity types and the QRT exception. Polynomial and
counting identities are checked for resonance orders 2 through 20 to index
100. The Laurent regression checks 456 valuations and leading coefficients
at two distinct precision cutoffs, 192 and 320, with precision propagated by
each operation. These finite checks supplement, not replace, the proof.

The package includes normal and optimized Python outputs and the portable-run
receipt. Corresponding equal-precision normal/optimized runs must agree byte
for byte. The two Laurent runs retain different precision metadata while
agreeing in every tested valuation and leading coefficient. General
positive-characteristic validity is not claimed from finite-field testing.

A development-only expected sign in one algebra check was corrected before
acceptance; the mathematical zero condition and manuscript argument were
unchanged. Both historical checker versions and outputs remain in the
research archive. The public portable package uses the corrected checker.

## Manuscript and package checks

The desktop LaTeX compiler succeeds. The exported PDF has eight pages with
no LaTeX errors, undefined references, missing citations, overfull or underfull
box warnings. All eight rendered pages were inspected; formulas, page breaks
and references are legible. A first local export was blocked by a missing
standard font download; a network-enabled rerun succeeded without changing
the source. That build failure is preserved in the research archive.

Only the author's name appears in the author line. Mercury Software GmbH,
email and GitHub are in its footnote. AI assistance is disclosed in ordinary
prose. Source ZIP membership and bytes are checked against a manifest before
release. The bibliography and source/novelty note identify substantial known
overlap. Independent review, formal verification and priority certification
remain false regardless of publication.